What is laplacetransform (t^2+1/2)^2 ?

The answer to laplacetransform (t^2+1/2)^2 is (24)/(s^5)+2/(s^3)+1/(4s)

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laplacetransform (t^2+1/2)^2

laplace transform

(t^{2} + \frac{1}{2})^{2}

\frac{24}{s ^{5}} + \frac{2}{s ^{3}} + \frac{1}{4 s}

Solution steps

L {(t^{2} + \frac{1}{2})^{2}}

Expand

(t^{2} + \frac{1}{2})^{2} : t^{4} + t^{2} + \frac{1}{4}

= L {t^{4} + t^{2} + \frac{1}{4}}

Use the linearity property of Laplace Transform:

For functions

f (t), g (t)

and constants

a, b : L {a \cdot f (t) + b \cdot g (t)} = a \cdot L {f (t)} + b \cdot L {g (t)}

= L {t^{4}} + L {t^{2}} + L {\frac{1}{4}}

L {t^{4}} : \frac{24}{s ^{5}}

L {t^{2}} : \frac{2}{s ^{3}}

L {\frac{1}{4}} : \frac{1}{4 s}

= \frac{24}{s ^{5}} + \frac{2}{s ^{3}} + \frac{1}{4 s}

(x^{2} + 2 y^{2}) d x - x y d y = 0

\frac{\partial}{\partial x} (x e^{\frac{s}{x}})

\int \frac{( 4 x ^{2} + 2 )}{( x ^{2} - 2 x + 2 ) ^{2}} d x

\frac{d}{d x} (g (x) (x)^{2})

y = 2 sin (x), at x = \frac{3}{4} π

What is laplacetransform (t^2+1/2)^2 ?
The answer to laplacetransform (t^2+1/2)^2 is (24)/(s^5)+2/(s^3)+1/(4s)